Calculus Examples

$6 x {(x^{2} - 1)}^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x {(x^{2} - 1)}^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x {(x^{2} - 1)}^{2}]$ .

$f' (x) = 6 \frac{d}{d x} (x {(x^{2} - 1)}^{2})$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x^{2} - 1)}^{2}$ .

$f' (x) = 6 (x \frac{d}{d x} ({(x^{2} - 1)}^{2}) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 1$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

$f' (x) = 6 (x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f' (x) = 6 (x (2 u \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.3.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

$f' (x) = 6 (x (2 (x^{2} - 1) \frac{d}{d x} (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$f' (x) = 6 (x (2 (x^{2} - 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1))) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 6 (x (2 (x^{2} - 1) (2 x + \frac{d}{d x} (- 1))) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f' (x) = 6 (x (2 (x^{2} - 1) (2 x + 0)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.4.4

Simplify the expression.

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Step 1.1.4.4.1

Add $2 x$ and $0$ .

$f' (x) = 6 (x (2 (x^{2} - 1) (2 x)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.4.4.2

Multiply $2$ by $2$ .

$f' (x) = 6 (x (4 (x^{2} - 1) x) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.5

Raise $x$ to the power of $1$ .

$f' (x) = 6 (x \cdot x (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.6

Raise $x$ to the power of $1$ .

$f' (x) = 6 (x \cdot x (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = 6 (x^{1 + 1} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.8

Add $1$ and $1$ .

$f' (x) = 6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} (x))$

Step 1.1.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \cdot 1)$

Step 1.1.10

Multiply ${(x^{2} - 1)}^{2}$ by $1$ .

$f' (x) = 6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2})$

Step 1.1.11

Simplify.

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Step 1.1.11.1

Apply the distributive property.

$f' (x) = 6 (x^{2} (4 x^{2} + 4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 1.1.11.2

Apply the distributive property.

$f' (x) = 6 (x^{2} (4 x^{2}) + x^{2} (4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 1.1.11.3

Apply the distributive property.

$f' (x) = 6 (x^{2} (4 x^{2})) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4

Combine terms.

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Step 1.1.11.4.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.11.4.1.1

Move $x^{2}$ .

$f' (x) = 6 (x^{2} x^{2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = 6 (x^{2 + 2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.1.3

Add $2$ and $2$ .

$f' (x) = 6 (x^{4} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.2

Move $4$ to the left of $x^{4}$ .

$f' (x) = 6 (4 \cdot x^{4}) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.3

Multiply $4$ by $6$ .

$f' (x) = 24 x^{4} + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.4

Multiply $4$ by $- 1$ .

$f' (x) = 24 x^{4} + 6 (x^{2} \cdot - 4) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.5

Move $- 4$ to the left of $x^{2}$ .

$f' (x) = 24 x^{4} + 6 (- 4 \cdot x^{2}) + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.4.6

Multiply $- 4$ by $6$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 {(x^{2} - 1)}^{2}$

Step 1.1.11.5

Simplify each term.

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Step 1.1.11.5.1

Rewrite ${(x^{2} - 1)}^{2}$ as $(x^{2} - 1) (x^{2} - 1)$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 ((x^{2} - 1) (x^{2} - 1))$

Step 1.1.11.5.2

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

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Step 1.1.11.5.2.1

Apply the distributive property.

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{2} (x^{2} - 1) - 1 (x^{2} - 1))$

Step 1.1.11.5.2.2

Apply the distributive property.

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1))$

Step 1.1.11.5.2.3

Apply the distributive property.

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3

Simplify and combine like terms.

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Step 1.1.11.5.3.1

Simplify each term.

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Step 1.1.11.5.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.11.5.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3.1.1.2

Add $2$ and $2$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3.1.2

Move $- 1$ to the left of $x^{2}$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3.1.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} - 1 \cdot - 1)$

Step 1.1.11.5.3.1.5

Multiply $- 1$ by $- 1$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} + 1)$

Step 1.1.11.5.3.2

Subtract $x^{2}$ from $- x^{2}$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 (x^{4} - 2 x^{2} + 1)$

Step 1.1.11.5.4

Apply the distributive property.

$f' (x) = 24 x^{4} - 24 x^{2} + 6 x^{4} + 6 (- 2 x^{2}) + 6 \cdot 1$

Step 1.1.11.5.5

Simplify.

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Step 1.1.11.5.5.1

Multiply $- 2$ by $6$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6 \cdot 1$

Step 1.1.11.5.5.2

Multiply $6$ by $1$ .

$f' (x) = 24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6$

Step 1.1.11.6

Add $24 x^{4}$ and $6 x^{4}$ .

$f' (x) = 30 x^{4} - 24 x^{2} - 12 x^{2} + 6$

Step 1.1.11.7

Subtract $12 x^{2}$ from $- 24 x^{2}$ .

$f' (x) = 30 x^{4} - 36 x^{2} + 6$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $30 x^{4} - 36 x^{2} + 6$ .

$30 x^{4} - 36 x^{2} + 6$

Step 2

Set the first derivative equal to $0$ then solve the equation $30 x^{4} - 36 x^{2} + 6 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$30 x^{4} - 36 x^{2} + 6 = 0$

Step 2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$30 u^{2} - 36 u + 6 = 0$

$u = x^{2}$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $6$ out of $30 u^{2} - 36 u + 6$ .

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Step 2.3.1.1

Factor $6$ out of $30 u^{2}$ .

$6 (5 u^{2}) - 36 u + 6 = 0$

Step 2.3.1.2

Factor $6$ out of $- 36 u$ .

$6 (5 u^{2}) + 6 (- 6 u) + 6 = 0$

Step 2.3.1.3

Factor $6$ out of $6$ .

$6 (5 u^{2}) + 6 (- 6 u) + 6 (1) = 0$

Step 2.3.1.4

Factor $6$ out of $6 (5 u^{2}) + 6 (- 6 u)$ .

$6 (5 u^{2} - 6 u) + 6 (1) = 0$

Step 2.3.1.5

Factor $6$ out of $6 (5 u^{2} - 6 u) + 6 (1)$ .

$6 (5 u^{2} - 6 u + 1) = 0$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor by grouping.

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Step 2.3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 1 = 5$ and whose sum is $b = - 6$ .

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Step 2.3.2.1.1.1

Factor $- 6$ out of $- 6 u$ .

$6 (5 u^{2} - 6 u + 1) = 0$

Step 2.3.2.1.1.2

Rewrite $- 6$ as $- 1$ plus $- 5$

$6 (5 u^{2} + (- 1 - 5) u + 1) = 0$

Step 2.3.2.1.1.3

Apply the distributive property.

$6 (5 u^{2} - 1 u - 5 u + 1) = 0$

Step 2.3.2.1.2

Factor out the greatest common factor from each group.

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Step 2.3.2.1.2.1

Group the first two terms and the last two terms.

$6 ((5 u^{2} - 1 u) - 5 u + 1) = 0$

Step 2.3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 (u (5 u - 1) - (5 u - 1)) = 0$

Step 2.3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 1$ .

$6 ((5 u - 1) (u - 1)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$6 (5 u - 1) (u - 1) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 u - 1 = 0$

$u - 1 = 0$

Step 2.5

Set $5 u - 1$ equal to $0$ and solve for $u$ .

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Step 2.5.1

Set $5 u - 1$ equal to $0$ .

$5 u - 1 = 0$

Step 2.5.2

Solve $5 u - 1 = 0$ for $u$ .

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Step 2.5.2.1

Add $1$ to both sides of the equation.

$5 u = 1$

Step 2.5.2.2

Divide each term in $5 u = 1$ by $5$ and simplify.

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Step 2.5.2.2.1

Divide each term in $5 u = 1$ by $5$ .

$\frac{5 u}{5} = \frac{1}{5}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 u}{5} = \frac{1}{5}$

Step 2.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{5}$

Step 2.6

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 2.6.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 2.6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 2.7

The final solution is all the values that make $6 (5 u - 1) (u - 1) = 0$ true.

$u = \frac{1}{5}, 1$

Step 2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1}{5}$

${(x^{2})}^{1} = 1$

Step 2.9

Solve the first equation for $x$ .

$x^{2} = \frac{1}{5}$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{5}}$

Step 2.10.2

Simplify $\pm \sqrt{\frac{1}{5}}$ .

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Step 2.10.2.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{5}}$

Step 2.10.2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{5}}$

Step 2.10.2.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.10.2.4

Combine and simplify the denominator.

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Step 2.10.2.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.10.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.10.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.10.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.10.2.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.10.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.10.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.10.2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.10.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.10.2.4.6.4

Cancel the common factor of $2$ .

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Step 2.10.2.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.10.2.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{5}}{5^{1}}$

Step 2.10.2.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{5}}{5}$

Step 2.10.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.10.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{5}}{5}$

Step 2.10.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{5}}{5}$

Step 2.10.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}$

Step 2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1$

Step 2.12

Solve the equation for $x$ .

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Step 2.12.1

Remove parentheses.

$x^{2} = 1$

Step 2.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 2.12.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.12.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.12.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.12.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.12.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 2.13

The solution to $30 x^{4} - 36 x^{2} + 6 = 0$ is $x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}, 1, - 1$ .

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}, 1, - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $6 x {(x^{2} - 1)}^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{\sqrt{5}}{5}$ .

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Step 4.1.1

Substitute $\frac{\sqrt{5}}{5}$ for $x$ .

$6 (\frac{\sqrt{5}}{5}) {({(\frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Combine $6$ and $\frac{\sqrt{5}}{5}$ .

$\frac{6 \sqrt{5}}{5} {({(\frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

Apply the product rule to $\frac{\sqrt{5}}{5}$ .

$\frac{6 \sqrt{5}}{5} {(\frac{{\sqrt{5}}^{2}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{6 \sqrt{5}}{5} {(\frac{{(5^{\frac{1}{2}})}^{2}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{1}{2} \cdot 2}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{2}{2}}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.2.4.1

Cancel the common factor.

$\frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{2}{2}}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2.4.2

Rewrite the expression.

$\frac{6 \sqrt{5}}{5} {(\frac{5^{1}}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.2.5

Evaluate the exponent.

$\frac{6 \sqrt{5}}{5} {(\frac{5}{5^{2}} - 1)}^{2}$

Step 4.1.2.2.3

Raise $5$ to the power of $2$ .

$\frac{6 \sqrt{5}}{5} {(\frac{5}{25} - 1)}^{2}$

Step 4.1.2.2.4

Cancel the common factor of $5$ and $25$ .

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Step 4.1.2.2.4.1

Factor $5$ out of $5$ .

$\frac{6 \sqrt{5}}{5} {(\frac{5 (1)}{25} - 1)}^{2}$

Step 4.1.2.2.4.2

Cancel the common factors.

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Step 4.1.2.2.4.2.1

Factor $5$ out of $25$ .

$\frac{6 \sqrt{5}}{5} {(\frac{5 \cdot 1}{5 \cdot 5} - 1)}^{2}$

Step 4.1.2.2.4.2.2

Cancel the common factor.

$\frac{6 \sqrt{5}}{5} {(\frac{5 \cdot 1}{5 \cdot 5} - 1)}^{2}$

Step 4.1.2.2.4.2.3

Rewrite the expression.

$\frac{6 \sqrt{5}}{5} {(\frac{1}{5} - 1)}^{2}$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{6 \sqrt{5}}{5} {(\frac{1}{5} - 1 \cdot \frac{5}{5})}^{2}$

Step 4.1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{6 \sqrt{5}}{5} {(\frac{1}{5} + \frac{- 1 \cdot 5}{5})}^{2}$

Step 4.1.2.5

Combine the numerators over the common denominator.

$\frac{6 \sqrt{5}}{5} {(\frac{1 - 1 \cdot 5}{5})}^{2}$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $5$ .

$\frac{6 \sqrt{5}}{5} {(\frac{1 - 5}{5})}^{2}$

Step 4.1.2.6.2

Subtract $5$ from $1$ .

$\frac{6 \sqrt{5}}{5} {(\frac{- 4}{5})}^{2}$

Step 4.1.2.7

Move the negative in front of the fraction.

$\frac{6 \sqrt{5}}{5} {(- \frac{4}{5})}^{2}$

Step 4.1.2.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.8.1

Apply the product rule to $- \frac{4}{5}$ .

$\frac{6 \sqrt{5}}{5} ({(- 1)}^{2} {(\frac{4}{5})}^{2})$

Step 4.1.2.8.2

Apply the product rule to $\frac{4}{5}$ .

$\frac{6 \sqrt{5}}{5} ({(- 1)}^{2} \frac{4^{2}}{5^{2}})$

Step 4.1.2.9

Simplify the expression.

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Step 4.1.2.9.1

Raise $- 1$ to the power of $2$ .

$\frac{6 \sqrt{5}}{5} (1 \frac{4^{2}}{5^{2}})$

Step 4.1.2.9.2

Multiply $\frac{4^{2}}{5^{2}}$ by $1$ .

$\frac{6 \sqrt{5}}{5} \cdot \frac{4^{2}}{5^{2}}$

Step 4.1.2.10

Combine.

$\frac{6 \sqrt{5} \cdot 4^{2}}{5 \cdot 5^{2}}$

Step 4.1.2.11

Multiply $5$ by $5^{2}$ by adding the exponents.

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Step 4.1.2.11.1

Multiply $5$ by $5^{2}$ .

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Step 4.1.2.11.1.1

Raise $5$ to the power of $1$ .

$\frac{6 \sqrt{5} \cdot 4^{2}}{5^{1} \cdot 5^{2}}$

Step 4.1.2.11.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 \sqrt{5} \cdot 4^{2}}{5^{1 + 2}}$

Step 4.1.2.11.2

Add $1$ and $2$ .

$\frac{6 \sqrt{5} \cdot 4^{2}}{5^{3}}$

Step 4.1.2.12

Simplify the numerator.

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Step 4.1.2.12.1

Raise $4$ to the power of $2$ .

$\frac{6 \sqrt{5} \cdot 16}{5^{3}}$

Step 4.1.2.12.2

Multiply $16$ by $6$ .

$\frac{96 \sqrt{5}}{5^{3}}$

Step 4.1.2.13

Raise $5$ to the power of $3$ .

$\frac{96 \sqrt{5}}{125}$

Step 4.2

Evaluate at $x = - \frac{\sqrt{5}}{5}$ .

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Step 4.2.1

Substitute $- \frac{\sqrt{5}}{5}$ for $x$ .

$6 (- \frac{\sqrt{5}}{5}) {({(- \frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $6 (- \frac{\sqrt{5}}{5})$ .

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Step 4.2.2.1.1

Multiply $- 1$ by $6$ .

$- 6 \frac{\sqrt{5}}{5} {({(- \frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.2.2.1.2

Combine $- 6$ and $\frac{\sqrt{5}}{5}$ .

$\frac{- 6 \sqrt{5}}{5} {({(- \frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.2.2.2

Move the negative in front of the fraction.

$- \frac{(6) \sqrt{5}}{5} {({(- \frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.2.2.3

Simplify each term.

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Step 4.2.2.3.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.3.1.1

Apply the product rule to $- \frac{\sqrt{5}}{5}$ .

$- \frac{6 \sqrt{5}}{5} {({(- 1)}^{2} {(\frac{\sqrt{5}}{5})}^{2} - 1)}^{2}$

Step 4.2.2.3.1.2

Apply the product rule to $\frac{\sqrt{5}}{5}$ .

$- \frac{6 \sqrt{5}}{5} {({(- 1)}^{2} \frac{{\sqrt{5}}^{2}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.2

Raise $- 1$ to the power of $2$ .

$- \frac{6 \sqrt{5}}{5} {(1 \frac{{\sqrt{5}}^{2}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.3

Multiply $\frac{{\sqrt{5}}^{2}}{5^{2}}$ by $1$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{{\sqrt{5}}^{2}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{{(5^{\frac{1}{2}})}^{2}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{1}{2} \cdot 2}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{2}{2}}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.3.4.4.1

Cancel the common factor.

$- \frac{6 \sqrt{5}}{5} {(\frac{5^{\frac{2}{2}}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4.4.2

Rewrite the expression.

$- \frac{6 \sqrt{5}}{5} {(\frac{5^{1}}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.4.5

Evaluate the exponent.

$- \frac{6 \sqrt{5}}{5} {(\frac{5}{5^{2}} - 1)}^{2}$

Step 4.2.2.3.5

Raise $5$ to the power of $2$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{5}{25} - 1)}^{2}$

Step 4.2.2.3.6

Cancel the common factor of $5$ and $25$ .

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Step 4.2.2.3.6.1

Factor $5$ out of $5$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{5 (1)}{25} - 1)}^{2}$

Step 4.2.2.3.6.2

Cancel the common factors.

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Step 4.2.2.3.6.2.1

Factor $5$ out of $25$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{5 \cdot 1}{5 \cdot 5} - 1)}^{2}$

Step 4.2.2.3.6.2.2

Cancel the common factor.

$- \frac{6 \sqrt{5}}{5} {(\frac{5 \cdot 1}{5 \cdot 5} - 1)}^{2}$

Step 4.2.2.3.6.2.3

Rewrite the expression.

$- \frac{6 \sqrt{5}}{5} {(\frac{1}{5} - 1)}^{2}$

Step 4.2.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{1}{5} - 1 \cdot \frac{5}{5})}^{2}$

Step 4.2.2.5

Combine $- 1$ and $\frac{5}{5}$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{1}{5} + \frac{- 1 \cdot 5}{5})}^{2}$

Step 4.2.2.6

Combine the numerators over the common denominator.

$- \frac{6 \sqrt{5}}{5} {(\frac{1 - 1 \cdot 5}{5})}^{2}$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $- 1$ by $5$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{1 - 5}{5})}^{2}$

Step 4.2.2.7.2

Subtract $5$ from $1$ .

$- \frac{6 \sqrt{5}}{5} {(\frac{- 4}{5})}^{2}$

Step 4.2.2.8

Move the negative in front of the fraction.

$- \frac{6 \sqrt{5}}{5} {(- \frac{4}{5})}^{2}$

Step 4.2.2.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.9.1

Apply the product rule to $- \frac{4}{5}$ .

$- \frac{6 \sqrt{5}}{5} ({(- 1)}^{2} {(\frac{4}{5})}^{2})$

Step 4.2.2.9.2

Apply the product rule to $\frac{4}{5}$ .

$- \frac{6 \sqrt{5}}{5} ({(- 1)}^{2} \frac{4^{2}}{5^{2}})$

Step 4.2.2.10

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.2.10.1

Move ${(- 1)}^{2}$ .

${(- 1)}^{2} \cdot - 1 \frac{6 \sqrt{5}}{5} \frac{4^{2}}{5^{2}}$

Step 4.2.2.10.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.2.2.10.2.1

Raise $- 1$ to the power of $1$ .

${(- 1)}^{2} \cdot {(- 1)}^{1} \frac{6 \sqrt{5}}{5} \frac{4^{2}}{5^{2}}$

Step 4.2.2.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- 1)}^{2 + 1} \frac{6 \sqrt{5}}{5} \frac{4^{2}}{5^{2}}$

Step 4.2.2.10.3

Add $2$ and $1$ .

${(- 1)}^{3} \frac{6 \sqrt{5}}{5} \frac{4^{2}}{5^{2}}$

Step 4.2.2.11

Raise $4$ to the power of $2$ .

${(- 1)}^{3} \frac{6 \sqrt{5}}{5} \frac{16}{5^{2}}$

Step 4.2.2.12

Raise $5$ to the power of $2$ .

${(- 1)}^{3} \frac{6 \sqrt{5}}{5} \frac{16}{25}$

Step 4.2.2.13

Raise $- 1$ to the power of $3$ .

$- \frac{6 \sqrt{5}}{5} \cdot \frac{16}{25}$

Step 4.2.2.14

Multiply $- \frac{6 \sqrt{5}}{5} \cdot \frac{16}{25}$ .

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Step 4.2.2.14.1

Multiply $\frac{16}{25}$ by $\frac{6 \sqrt{5}}{5}$ .

$- \frac{16 (6 \sqrt{5})}{25 \cdot 5}$

Step 4.2.2.14.2

Multiply $6$ by $16$ .

$- \frac{96 \sqrt{5}}{25 \cdot 5}$

Step 4.2.2.14.3

Multiply $25$ by $5$ .

$- \frac{96 \sqrt{5}}{125}$

Step 4.3

Evaluate at $x = 1$ .

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Step 4.3.1

Substitute $1$ for $x$ .

$6 (1) {({(1)}^{2} - 1)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $6$ by $1$ .

$6 {({(1)}^{2} - 1)}^{2}$

Step 4.3.2.2

One to any power is one.

$6 {(1 - 1)}^{2}$

Step 4.3.2.3

Subtract $1$ from $1$ .

$6 \cdot 0^{2}$

Step 4.3.2.4

Raising $0$ to any positive power yields $0$ .

$6 \cdot 0$

Step 4.3.2.5

Multiply $6$ by $0$ .

$0$

Step 4.4

Evaluate at $x = - 1$ .

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Step 4.4.1

Substitute $- 1$ for $x$ .

$6 (- 1) {({(- 1)}^{2} - 1)}^{2}$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Multiply $6$ by $- 1$ .

$- 6 {({(- 1)}^{2} - 1)}^{2}$

Step 4.4.2.2

Raise $- 1$ to the power of $2$ .

$- 6 {(1 - 1)}^{2}$

Step 4.4.2.3

Subtract $1$ from $1$ .

$- 6 \cdot 0^{2}$

Step 4.4.2.4

Raising $0$ to any positive power yields $0$ .

$- 6 \cdot 0$

Step 4.4.2.5

Multiply $- 6$ by $0$ .

$0$

Step 4.5

List all of the points.

$(\frac{\sqrt{5}}{5}, \frac{96 \sqrt{5}}{125}), (- \frac{\sqrt{5}}{5}, - \frac{96 \sqrt{5}}{125}), (1, 0), (- 1, 0)$

Step 5